Discrete Energy on Rectifiable Sets

by Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff

Overview

This book aims to provide an introduction to the broad and dynamic subject of discrete energy problems and point configurations. Written by leading authorities on the topic, this treatise is designed with the graduate student and further explorers in mind. The presentation includes a chapter of preliminaries and an extensive Appendix that augments a course in Real Analysis and makes the text self-contained. Along with numerous attractive full-color images, the exposition conveys the beauty of the subject and its connection to several branches of mathematics, computational methods, and physical/biological applications.

This work is destined to be a valuable research resource for such topics as packing and covering problems, generalizations of the famous Thomson Problem, and classical potential theory in R^d. It features three chapters dealing with point distributions on the sphere, including an extensive treatment of Delsarte–Yudin–Levenshtein linearprogramming methods for lower bounding energy, a thorough treatment of Cohn–Kumar universality, and a comparison of 'popular methods' for uniformly distributing points on the two-dimensional sphere. Some unique features of the work are its treatment of Gauss-type kernels for periodic energy problems, its asymptotic analysis of minimizing point configurations for non-integrable Riesz potentials (the so-called Poppy-seed bagel theorems), its applications to the generation of non-structured grids of prescribed densities, and its closing chapter on optimal discrete measures for Chebyshev (polarization) problems.

Product Details

ISBN-13:	9780387848075
Publisher:	Springer New York
Publication date:	11/01/2019
Series:	Springer Monographs in Mathematics
Edition description:	1st ed. 2019
Pages:	666
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Sergiy V. Borodachov is a Professor of Mathematics at Towson University, which he joined in 2008. Prof. Borodachov's primary research interests include approximation theory, numerical analysis, and minimal energy problems. He authored or co-authored more than 30 research articles and gave more than 90 talks at research conferences and seminars.

Douglas P. Hardin is a Professor of Mathematics and a Professor of Biomedical Informatics at Vanderbilt University. His research interests include discrete minimum energy problems, fractals, harmonic analysis (wavelets), inverse problems, and machine learning. Hardin has authored or co-authored over 115 research publications, 2 monographs, and co-edited 3 research journal special issues.

Edward B. Saff is a Professor of Mathematics at Vanderbilt University and Director of the Center for Constructive Approximation. His research areas include approximation theory, numerical analysis, and potential theory. Saff is a Fellow of the American Mathematics Society, a Foreign Member of the Bulgarian Academy of Science, and was a recipient of both a Guggenheim and a Fulbright Fellowships. He has authored or co-authored over 270 research articles, 4 research monographs and 4 textbooks, and is co-Editor-in-Chief and Managing Editor of the research journal Constructive Approximation. Prof. Saff also serves on the boards of 3 other research journals.

Table of Contents

0. An Overview: Discretizing Manifolds via Particle Interactions.-1. Preliminaries.- 2. Basics of Minimal Energy.- 3.-Introduction to Packing and Covering.- 4. Continuous and Discrete Energy.- 5. LP Bounds on the Sphere.- 6. Asymptotics for Energy Minimizing Congurations on Sd.- 7. Some Popular Algorithms for Distributing Points on S2.- 8. Minimal Energy in the Hypersingular Case.- 9. Minimal Energy Asymptotics in the "Harmonic Series" Case.- 10. Periodic Riesz Energy.- 11. Congurations with non-Uniform Distribution.- 12. Low Complexity Energy Methods for Discretization.- 13. Best-Packing on Compact Sets.- 14. Optimal Discrete Measures for Potentials: Polarization (Chebyshev) Constants.- Appendix.- References.- List of Symbols.- Index.

Editorial Reviews

The authors have done an excellent job at completeness in showcasing the range of techniques one can bring to such a seemingly simple subject. This is a book for every practicing mathematician who is interested in the subject. I am happy it has a place on my bookshelf.” (Jeff Ibbotson, MAA Reviews, July 4, 2021)

“The book is mostly self-contained and has been written with graduate students in mind. … The reviewer certainly thinks that this book has the potential to become a standard text for students new to the rich and vibrant area of discrete energy problems and point configurations and a valuable resource for researchers.” (Johann S. Brauchart, Mathematical Reviews, April, 2021)

“The authors have done an excellent work by taking the reader, who is primarily supposed to be a graduate student, from the basics of Real Analysis to the frontiers of research on several mathematical topics, what turns the text of interest for both students and research professionals. The vast content of the book will certainly provide the reader with an extremely valuable source on this fascinating subject.” (Antonio Roberto da Silva, zbMATH 1437.41002, 2020)

From the Publisher